Kazhdan's orthogonality conjecture for real reductive groups

TL;DR

This paper generalizes Harish-Chandra's character orthogonality relations from discrete series to all Harish-Chandra modules for real reductive groups, extending Kazhdan's conjecture from p-adic to real groups.

Contribution

It proves a broad generalization of Harish-Chandra's orthogonality relations, confirming Kazhdan's conjecture for real reductive groups.

Findings

Established orthogonality relations for all Harish-Chandra modules

Extended Kazhdan's conjecture from p-adic to real groups

Provided new tools for representation theory of real reductive groups

Abstract

We prove a generalization of Harish-Chandra's character orthogonality relations for discrete series to arbitrary Harish-Chandra modules for real reductive Lie groups. This result is an analogue of a conjecture by Kazhdan for $\mathfrak p$-adic reductive groups proved by Bezrukavnikov, and Schneider and Stuhler.